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Chapter 19: Problem 1888

If \(\cos ^{-1} x-\sin ^{-1} x=(\pi / 4)\) then \(x=\) (a) \([\sqrt{\\{} 2-\sqrt{2}\\} / 2]\) (b) \([\sqrt{\\{2}+\sqrt{2}\\} / 2]\) (c) \(\sqrt{2}-1\) (d) \(\sqrt{2}+1\)

Short Answer

Expert verified
The short answer is: \(x = [\sqrt{2 - \sqrt{2}} / 2]\) (Option a).

Step by step solution

01

Since we know that \(\sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2}\), we can substitute \(\sin^{-1}(x)\) with the expression in terms of \(\cos^{-1}(x)\). So, \(\sin^{-1}(x) = \frac{\pi}{2} - \cos^{-1}(x)\) Now, we can replace \(\sin^{-1}(x)\) in the given equation: \(\cos^{-1}(x) - (\frac{\pi}{2} - \cos^{-1}(x)) = \frac{\pi}{4}\) #Step 2: Solve for cos inverse x#

Simplify the equation to find the value of \(\cos^{-1}(x)\): \(2\cos^{-1}(x) - \frac{\pi}{2}= \frac{\pi}{4}\) Now, add \(\frac{\pi}{2}\) to both sides and divide by 2: \(\cos^{-1}(x) = \frac{\pi}{4} + \frac{\pi}{8} = \frac{3\pi}{8}\) #Step 3: Find x using cosine inverse#
02

Now, we can find the value of \(x\) by taking the cosine of both sides of the equation: \(x = \cos(\frac{3\pi}{8})\) Now we can use the half-angle identity to find the value of the cosine function: \(\cos(\frac{3\pi}{8}) = \sqrt{\frac{1 + \cos(\frac{3\pi}{4})}{2}}\) Since \(\cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2}\), we can replace it in the equation: \(x = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}\) #Step 4: Simplify the expression#

Now, we just need to simplify the expression for \(x\): \(x = \sqrt{\frac{2 - \sqrt{2}}{4}}\) Upon comparing the expression to options (a), (b), (c), and (d), we find the answer is: \(x = [\sqrt{2 - \sqrt{2}} / 2]\) (Option a).

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Most popular questions from this chapter

\(\sin \left[\cot ^{-1}\left(\cos \left(\tan ^{-1} x\right)\right)\right]=\) (b) \(\left.\sqrt{[}\left(x^{2}+1\right) /\left(x^{2}+2\right)\right]\) (c) \(\left[x / \sqrt{ \left.\left(x^{2}+2\right)\right]}\right.\) (d) \(\left[1 / \sqrt{ \left.\left(x^{2}+2\right)\right]}\right.\)

If \(4 \sin ^{-1} x+3 \cos ^{-1} x=2 \pi\), then \(x=\) (a) 1 (b) \(-1\) (c) \((1 / 2)\) (d) \(-(1 / 2)\)

The number of solution of \(\cos x+\cos 2 x+\cos 3 x+\cos 4 x=0\) \(\mathrm{x} \in[0,2 \pi]\) is (a) 4 (b) 5 (c) 6 (d) 7

\(A=\tan \left[\sin ^{-1}(3 / 5)+\cot ^{-1}(3 / 2)\right]\) then \(A=\) (a) \((17 / 2)\) (b) \((17 / 6)\) (c) \((17 / 12)\) (d) \((6 / 17)\)

\(\tan ^{-1}(\tan 4)-\tan ^{-1}(\tan (-6))+\cos ^{-1}(\cos 10)=\) (a) 16 (b) \(\pi\) (c) \(-\pi\) (d) \(5 \pi-12\)
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